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A321662 Number of non-isomorphic multiset partitions of weight n whose incidence matrix has all distinct entries. 6
1, 1, 1, 3, 3, 5, 13, 15, 23, 33, 49, 59, 83, 101, 133, 281, 321, 477, 655, 941, 1249, 1795, 2241, 3039, 3867, 5047, 6257, 8063, 11459, 13891, 18165, 23149, 29975, 37885, 49197, 61829, 89877, 109165, 145673, 185671, 246131, 310325, 408799, 514485, 668017, 871383 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The incidence matrix of a multiset partition has entry (i, j) equal to the multiplicity of vertex i in part j.

Also the number of positive integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, with all different entries.

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = Sum_{k>=1} (A121860(k) + A121860(k+1) - 2)*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

EXAMPLE

Non-isomorphic representatives of the a(3) = 3 through a(7) = 15 multiset partitions:

  {{111}}    {{1111}}    {{11111}}    {{111111}}      {{1111111}}

  {{122}}    {{1222}}    {{11222}}    {{112222}}      {{1112222}}

  {{1}{11}}  {{1}{111}}  {{12222}}    {{122222}}      {{1122222}}

                         {{1}{1111}}  {{122333}}      {{1222222}}

                         {{11}{111}}  {{1}{11111}}    {{1223333}}

                                      {{11}{1111}}    {{1}{111111}}

                                      {{1}{11222}}    {{11}{11111}}

                                      {{11}{1222}}    {{111}{1111}}

                                      {{112}{222}}    {{1}{112222}}

                                      {{122}{222}}    {{11}{12222}}

                                      {{2}{11222}}    {{112}{2222}}

                                      {{22}{1222}}    {{122}{2222}}

                                      {{1}{11}{111}}  {{2}{112222}}

                                                      {{22}{12222}}

                                                      {{1}{11}{1111}}

MATHEMATICA

(* b = A121860 *) b[n_] := Sum[n!/(d! (n/d)!), {d, Divisors[n]}];

(* c = A008289 *) c[n_, k_] := c[n, k] = If[n < k || k < 1, 0, If[n == 1, 1, c[n - k, k] + c[n - k, k - 1]]];

a[n_] := If[n == 0, 1, Sum[ (b[k] + b[k + 1] - 2) c[n, k], {k, 1, n}]];

a /@ Range[0, 45] (* Jean-François Alcover, Sep 14 2019 *)

PROG

(PARI) \\ here b(n) is A121860(n).

b(n)={sumdiv(n, d, n!/(d!*(n/d)!))}

seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, b(n-1)+b(n)-2))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

CROSSREFS

Cf. A000219, A007716, A008289, A059201, A114736, A117433, A120733, A121860, A321653, A321659, A321660, A321661.

Sequence in context: A079439 A231895 A218426 * A320176 A298478 A144419

Adjacent sequences:  A321659 A321660 A321661 * A321663 A321664 A321665

KEYWORD

nonn

AUTHOR

Gus Wiseman, Nov 15 2018

EXTENSIONS

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018

STATUS

approved

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Last modified November 28 13:42 EST 2021. Contains 349412 sequences. (Running on oeis4.)